If I learn Nigerian, wear Nigerian dress, cook Nigerian food and go and live in Nigeria, can I declare that I am, in fact, Nigerian and always have been on the inside, even though outwardly I am a white Yorkshirewoman?
Well, yes. If you mean yoruba or something instead of 'Nigerian' which isn't a language. In the same way that my husband, who was born in Asia, is Scottish, because he moved to Scotland, became fluent in English, married a Scotswoman and assimilated into the culture. Or are you saying he'll always be Asian because only white people with Scots ancestry can be Scottish? Which I don't think you mean to say, but it's actually a good analogy to the trans woman conversation.
If I say 'yes, you could be', that's because I'm defining 'Nigerian' as a culture, a nationality, a way of life. You're defining it as skin colour. In the same way, some people define 'woman' as 'genital configuration observed at birth'. It's a biological thing for them, much as 'Nigerian' is, I guess, a biological thing for you. If, on the other hand, you define 'woman' as a social role, a construct with elements of socialised behaviour, societal role, style of interaction, visual presentation, personal identity etc, then of course someone can become a woman.
It's about how we define things, and definitions are always complex and subjective, in science and in all things.